PHYSICS OF PLASMASVOLUME 8, NUMBER 6

JUNE 2001

Stabilization effect of magnetic shear on the diocotron instability
Shigeo Kondoh, Tomoya Tatsuno, and Zensho Yoshida
Graduate School of Frontier Sciences, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan

共Received 13 December 2000; accepted 7 February 2001兲
The diocotron instability in a magnetized non-neutral plasma is a close cousin of the Kelvin–
Helmholtz instability. A sheared magnetic field brings about coupling between the diocotron modes
and the Langmuir waves that propagate along the magnetic field. The motion of electrons parallelto the magnetic field cancels the electric charge produced by the diocotron modes, resulting instabilization of the diocotron instability. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1371521兴

I. INTRODUCTION

shear analytically. The diocotron instability is formallyequivalent to the Kelvin–Helmholtz instabilities in fluids andplasmas.11 The magnetic shear stabilization of these instabilities is of the common interest and has a variety of applications 共see Sec. IV兲.

Recently a variety of new concepts on non-neutral
plasma confinement have been proposed,1–3 which significantly differ from the conventional Penning/Malmberg
trap.4,5 The Prototype Ring Trap 共Proto-RT兲 experiment1,6,7
is aimed at pure magnetic confinement of a toroidal nonneutral plasma that is not in a rigid-rotating thermal equilibrium state. In such a system, the plasma flow is generally
sheared, and the diocotron instability8 can be destabilized.
The application of magnetic shear is expected to be mosteffective to stabilize the electrostatic modes. However, an
exact stability analysis has not been completed, except forthe special case of an electron beam with a relativistic
speed.9
The physical mechanism of the diocotron instability is
explained as follows10 共see Fig. 1兲: When a non-neutral slab
plasma has a finite thickness, a perturbation on one of thetwo plasma surfaces produces surface charges. The resulting
perturbed electric field yields an EÃB flow in the plasma,
and the opposite surface is also perturbed. The motion of the
opposite surface brings about a reciprocal perturbation, andthe waves on the two surfaces couple with each other. Undercertain conditions, this coupling yields a positive feedback,and the diocotron instability occurs.
When a non-neutral plasma is confined in a uniform
magnetic field, the diocotron modes propagating in the perpendicular direction to the magnetic field are independent ofany modes that propagate in the parallel direction. However,if the magnetic field has a shear 共see Fig. 2兲, the wave vector
may have a local parallel component k 储 (x), and the diocotron modes interact with the parallel modes, such as theLangmuir wave or the plasma oscillation. In a cold nonneutral plasma, the surface charge perturbation produced by
the diocotron modes is short-circuited by the parallel motionof charged particles, if the diocotron frequency ␻ D is muchsmaller than the plasma frequency ␻ p , i.e., when a low density plasma embedded in a strong magnetic field. Therefore,
we expect that the diocotron instability is stabilized in asheared magnetic field.
In this paper, we consider a slab plasma with a flat-top
density profile and show the stabilizing effect of magnetic
1070-664X/2001/8(6)/2635/6/$18.00

II. EIGENEQUATION FOR DIOCOTRON MODES IN A
SHEARED MAGNETIC FIELD
A. Slab plasma model in a sheared magnetic field

We consider a slab electron plasma embedded in asheared magnetic field 共see Fig. 2兲. The plasma has a finite
thickness 2⌬ in the x-direction. We assume that all equilibrium quantities are functions of only x. We consider asheared magnetic field such asB⫽„0,B y 共 x 兲 ,B z …,

共1兲

where B z is a constant. Since a non-neutral plasma has aself-electric field, there is a stationary flow that is approximately equivalent to the EÃB drift for low densities.
The governing equations are⳵n⫹v•“n⫹n“•v⫽0,
⳵t

共2兲

⳵v
1
⫹ 共 v•“ 兲 v⫽⫺ 2 共 E⫹vÃB兲 ,
⳵t
s

共3兲

ⵜ 2 ␾ ⫽⫺n,共4兲where n is normalized by the typical electron density n 0 (0),⫺1t by the inverse of the diocotron frequency ␻ D⫽␧ 0 B z /en 0 (0) 共␧ 0 is the vacuum dielectric constant and eis the elementary electric charge兲, the spatial coordinates x,
y, z by the half thickness of the slab plasma ⌬, v by themean flow velocity at the plasma surface 兩 v0 (1) 兩 , B by theaxial magnetic field B z , E by the mean electric field at theplasma surface 兩 E(1) 兩 , s⬅ ␻ p / ␻ c ⫽ ␻ D / ␻ p is a dimensionless parameter, and E⫽⫺“ ␾ . Since we consider a lowdensity plasma in a strong axial magnetic field, we may assume sⰆ1. In this limit, we can replace Eq. 共3兲 with
v⬜ ⫽
2635

⫺“ ␾ ÃB共 x 兲,
B共 x 兲2

共5兲
© 2001 American Institute of Physics

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2636

Phys. Plasmas, Vol. 8, No. 6, June 2001

Kondoh, Tatsuno, and Yoshidax⫽x ⬘ ,y⫽Bz
Byy ⬘⫹ z ⬘,
B
B

z⫽⫺

共9兲

Bz
By
y ⬘⫹ z ⬘,BBon Eqs. 共7兲. The new z ⬘ -axis is set to parallel the direction of
the magnetic field, and the new y ⬘ -axis is perpendicular toboth the x- and z ⬘ -directions. Equation 共7兲 now reads as„␻ ⫺k⬜ 共 x 兲v⬜0 …n 1 ⫹FIG. 1. A physical picture of diocotron modes in a uniform magnetic field.
Perturbation on one of the two plasma surfaces produces the surface charge
and causes the electrostatic field perturbation. This perturbed electric fieldshakes the body of the plasma itself through E⫻B drift, and the othersurface is also perturbed. The perturbation on the latter surface in turnshakes the former one in the same way. Thus, the waves on the two surfaces
couple with each other. Under certain conditions, the diocotron modes canbe unstable.

1⳵v储
⫹ 共 v•“ 兲v 储 ⫽ 2 “ 储 ␾ ,⳵ts

共 ␻ ⫺k y v 0y ⫺k z v 0z 兲 n 1 ⫺冋k⬜ 共 x 兲 ⫽k 储共 x 兲⫽

where we have Fourier-transformed all perturbed variables⌿(x,y,z,t) as⌿ 共 x,y,z,t 兲 ⫽⌿ 0 共 x 兲 ⫹⌿ 1 共 x 兲 exp关 i 共 ␻ t⫺k y y⫺k z z 兲兴 .

共8兲冑1⫹B 2yk y B y ⫹k z

冑1⫹B 2y

,

共11兲

.

共12兲

d 2␾ 1⫺k 2 ␾ 1 ⫽n 1 ,
dx 2

册

共7兲

k y ⫺k z B ySimilarly, from Eqs. 共4兲 and 共6兲, we obtaind B y k z ⫺B z k y
n 0 ␾ 1 ⫺in 0 “•v储 1
dx
B2

⫽0,

冊

where k⬜ (x) and k 储 (x) are the wave vector, in the perpendicular and parallel directions, with respect to the magnetic
field, respectively;

共6兲

where B(x)⫽ 冑B y (x) 2 ⫹B z2 . We are neglecting the polarization drift velocity vp , because vp ⫽O(s 2 )Ⰶ1. Substituting
Eq. 共5兲 into Eq. 共2兲 and linearizing the resulting equation, we
obtain冉d k⬜ 共 x 兲 n 0
␾ 1 ⫺n 0 k 储 共 x 兲v 储 1 ⫽0,
dx
B
共10兲

共 ␻ ⫺k⬜ v⬜0 兲v 储 1 ⫽⫺

共13兲
k储
␾ ,
s2 1

共14兲

where k 2 ⬅k⬜ (x) 2 ⫹k 储 (x) 2 ⫽k 2y ⫹k z2 , which is independentof x. Substituting Eqs. 共13兲 and 共14兲 into Eq. 共10兲, we obtain
the eigenequation for the perturbed electrostatic potential as

冉

冊

冉 冊

d 2␾ 1d k⬜ n 0
1⫺k 2 ␾ 1 ⫹␾1dx 2
␻ ⫺k⬜ v⬜0 dx B

Here we perform a coordinate transform,n 0 k 2储⫹ 2
␾ ⫽0.
s 共 ␻ ⫺k⬜ v⬜0 兲 2 1

共15兲

B. Coupling between diocotron wave and plasma
oscillation

The last term of Eq. 共15兲 represents the coupling of theparallel dynamics and the diocotron modes. In the limit of ashearless magnetic field (B y ⫽0), the diocotron modes andplasma oscillation are decoupled, and, hence, the last term of
the eigenequation 共15兲 vanishes (k 储 ⫽0), and 共15兲 reduces to冉FIG. 2. A physical picture of stabilizing effect of a sheared magnetic field
on diocotron modes. The wave vector almost always has a local parallel
component, and the diocotron modes cannot be independent of the parallelmodes, such as the Langmuir wave or the plasma oscillation. Therefore, acoupling between them is caused. In a cold non-neutral plasma, the surface
charge perturbation produced by the diocotron modes is canceled by theparallel motion of charged particles. Thus, the diocotron instability is stabilized by the sheared magnetic field.

冊

d 2␾ 1dn 0
ky⫺k 2y ␾ 1 ⫹
␾ ⫽0,
dx 2
␻ ⫺k y v y0 dx 1

共16兲

which is equivalent to Rayleigh’s equation for the Kelvin–
Helmholtz instability.12,13 The diocotron instability, which isthus a cousin of the Kelvin–Helmholtz instability, is causedby the non-self-adjointness brought about by the second termin Eq. 共15兲.
In the sheared magnetic field, the diocotron modes always couple with the plasma oscillation in the direction parallel to the magnetic field. This interaction is represented by

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Phys. Plasmas, Vol. 8, No. 6, June 2001

Stabilization effect of magnetic shear . . .

the last term in Eq. 共15兲. The last term includes a small
parameter s 2 in its denominator, so a significant change isbrought about in the characteristics of the equation. Physically this means that the parallel motion of the plasma easilycancels the charge perturbation and has a strong shortcircuiting effect under the condition that the time scale of theplasma oscillation is much smaller than the diocotron modes(s⫽ ␻ D / ␻ p Ⰶ1).
In the next section, we will show analytically that thelast term has a stabilizing effect.

A. Nonresonant frequency regime

First, we show that the diocotron modes are stabilized( ␻ i ⫽0) for wave numbers without resonance between thephase velocity and the plasma flow, that is, ␻ r ⫺k⬜ v⬜0 ⫽0for all x. Multiplying Eq. 共15兲 by ␾ * and integrating it over(⫺⬁,⬁), we obtain from the imaginary part,

␻i

冕 冋␻
⬁

冉 冊
册

2n 0 k 2储 共 ␻ r ⫺k⬜ v⬜0 兲
s 2 兩 ␻ ⫺k⬜ v⬜0 兩 4

兩 ␾ 1 兩 2 dx⫽0.

n 0共 x 兲 k 储共 x 兲 2⬅a 2 ⫽constⰇ1,
s2

共20兲

k⬜ 共 x 兲v⬜0 共 x 兲 ⫽x.

共21兲

Under these assumptions, Eq. 共15兲 reduces toa2d 2␾ 1⫹
␾ ⫽0dx 2
共 ␻ ⫺x 兲 2 1

共22兲

共 兩 x 兩 ⬎1 兲 ,共23兲with the boundary condition, Eq. 共18兲, and the jump conditionsd␾1
f 共 ⫺1 兲d␾1
␾ 共 ⫺1 兲 ,共 ⫺1⫹ ⑀ 兲 ⫺共 ⫺1⫺ ⑀ 兲 ⫽⫺dx
dx
␻ ⫹1 1

共17兲

共24兲
共25兲

The general solution to Eqs. 共22兲 and 共23兲 is givenby (k⬎0)

Here we used the boundary condition␾ 1 共 ⫾⬁ 兲 ⫽0.

共 兩 x 兩 ⬍1 兲 ,

d␾1
f 共1兲
d␾1
␾ 共 1 兲.
共 1⫹ ⑀ 兲 ⫺共 1⫺ ⑀ 兲 ⫽dxdx
␻ ⫺1 11
d k⬜ n 0
B⫺⬁ 兩 ⫺k⬜ v⬜0 兩 dx⫹

where f (x) is a given finite function. Furthermore, we assume

d 2␾ 1⫺k 2 ␾ 1 ⫽0dx 2III. STABILIZING EFFECT DUE TO PARALLEL
MOTION

2637

共18兲

冦

␾ I⫽C 1 e kx

共 x⬍⫺1 兲 ,冑2␾ II⫽C 2 共 ␻ ⫺x 兲 (1⫺ 1⫺4a )/2
冑2⫹C 3 共 ␻ ⫺x 兲 (1⫹ 1⫺4a )/2

Since s 2 Ⰶ1 and ␻ r ⫺k⬜ v⬜0 ⫽0 at any point in the plasma
region, we obtain ␻ i ⫽0, which means stability. This mathematical treatment is the same as the standard Rayleigh’s
analysis.12

␾ 1共 x 兲 ⫽B. Dispersion relation with resonances

under the boundary condition 共18兲. Substituting Eq. 共26兲 into
the jump conditions 共24兲 and 共25兲 gives

If the plasma has a resonant point, the analysis in Sec.
III A does not apply to check whether the eigenvalues ␻ for
Eq. 共15兲 are real or not. In this case, we need to solve Eq.
共15兲 directly. The eigenfunction determined by Eq. 共15兲 isoscillatory, because the sign of the last term, which we assumed to be very large (sⰆ1), is positive. If ␻ is not real,
the real and imaginary parts of the eigenfunction have a relative phase angle of about ␲ /2. When we consider a density
profile with a sharp boundary, we have to connect both thereal and imaginary parts of the eigenfunction at the plasma
surfaces using the same boundary condition. If both of them
have a different phase angle, this process fails, which implies
that ␻ must be real.
The essential characteristic of this eigenvalue problem iswell understood by the following simplified model. First, we
neglect the second term k 2 ␾ 1 in the bracket of the first termof Eq. 共15兲 in the plasma region (⫺1,1), since it is muchsmaller than the last term when n 0 ⯝1. We also assume thatk⬜ n 0 /B jumps at x⫽⫾1 and its variation is negligible anywhere else, i.e.,冉 冊d k⬜ n 0⫽ f 共 x 兲关 ␦ 共 x⫹1 兲 ⫺ ␦ 共 x⫺1 兲兴 ,
dx B

共19兲

⫹C 3 共 ␻ ⫺x 兲 (1⫹2ai)/2

␾ III⫽C 4 e

冋冉 冊

C2 ⫺

⫺kx

共 兩 x 兩 ⬍1 兲 ,共 x⬎1 兲 ,

冉1
f 共 ⫺1 兲⫺ai 共 ␻ ⫹1 兲 ⫺(1⫹2ai)/2⫹
⫺k
2
␻ ⫹1

册 冋冉 冊冉冊⫻ 共 ␻ ⫹1 兲 (1⫺2ai)/2 ⫹C 3 ⫺⫻ 共 ␻ ⫹1 兲 (⫺1⫹2ai)/2⫹

冋冉 冊

C2 ⫺

共26兲

⯝C 2 共 ␻ ⫺x 兲 (1⫺2ai)/2

1
⫹ai
2

册

f 共 ⫺1 兲⫺k 共 ␻ ⫹1 兲 (1⫹2ai)/2 ⫽0,
␻ ⫹1

冉

1
f 共1兲
⫺ai 共 ␻ ⫺1 兲 ⫺(1⫹2ai)/2⫹
⫹k
2
␻ ⫺1

册 冋冉 冊冉 冊⫻ 共 ␻ ⫺1 兲 (1⫺2ai)/2 ⫹C 3 ⫺⫻ 共 ␻ ⫺1 兲 (⫺1⫹2ai)/2⫹

冊

1
⫹ai
2

共27兲

冊

册

f 共1兲
⫹k 共 ␻ ⫺1 兲 (1⫹2ai)/2 ⫽0.
␻ ⫺1
共28兲

If Eqs. 共27兲 and 共28兲 have nontrivial solutions for C 2 andC 3 , the following relation must be satisfied:Downloaded 24 May 2001 to 130.69.86.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

2638

Phys. Plasmas, Vol. 8, No. 6, June 2001

冋冉 冊
冋冉 冊
冉
冊
⫺

冉

Kondoh, Tatsuno, and Yoshida

冊

1
f 共 ⫺1 兲⫺ai 共 ␻ ⫹1 兲 ⫺(1⫹2ai)/2⫹⫺k 共 ␻ ⫹1 兲 (1⫺2ai)/2
2
␻ ⫹1

⫻ ⫺

冉 冊
册冋冉 冊

册

册冋冉 冊
冉 冊

1
f 共1兲
1
⫹ai 共 ␻ ⫺1 兲 (⫺1⫹2ai)/2⫹⫹k 共 ␻ ⫺1 兲 (1⫹2ai)/2 ⫺ ⫺ ⫹ai 共 ␻ ⫹1 兲 (⫺1⫹2ai)/2
2
␻ ⫺1
2

册

f 共 ⫺1 兲1
f 共1兲
⫺k 共 ␻ ⫹1 兲 (1⫹2ai)/2 ⫻ ⫺ ⫺ai 共 ␻ ⫺1 兲 ⫺(1⫹2ai)/2⫹⫹k 共 ␻ ⫺1 兲 (1⫺2ai)/2 ⫽0.
␻ ⫹1
2
␻ ⫺1

⫹

共29兲

Equation 共29兲 is the dispersion relation. We can show that ␻ is real for Eq. 共29兲. Substituting ␻ ⫽ ␻ r ⫹i ␻ i into Eq. 共29兲, we
obtain冋

共 a⫹k ␻ i 兲 ⫺k ␻ r2 ⫹
2

2

冉 冊

冉 冊

1 2
1
k⫹
⫹ f 共 ⫺1 兲 f 共 1 兲 ⫺ k⫹ „f 共 ⫺1 兲 ⫹ f 共 1 兲 …
22册⫹k ␻ r „f 共 ⫺1 兲 ⫺ f 共 1 兲 …⫹i 共 a⫹k ␻ i 兲 „f 共 ⫺1 兲 ⫺ f 共 1 兲 ⫺2k ␻ r … 共 ␻ ⫹1 兲 2ai

冉 冊冋

⫽ 共 a⫺k ␻ i 兲 2 ⫺k 2 ␻ r2 ⫹ k⫹

冉 冊

1 2
1
⫹ f 共 ⫺1 兲 f 共 1 兲 ⫺ k⫹ „f 共 ⫺1 兲 ⫹ f 共 1 兲 …
22册⫹k ␻ r „f 共 ⫺1 兲 ⫺ f 共 1 兲 …⫺i 共 a⫺k ␻ i 兲 „f 共 ⫺1 兲 ⫺ f 共 1 兲 ⫺2k ␻ r … 共 ␻ ⫺1 兲 2ai ,where ␻ r ⫽Re ␻ and ␻ i ⫽Im ␻. Taking the absolute numberof Eq. 共30兲 gives冋 冉 冊册

兩 A 1兩
␻ ⫹1⫽exp 2a arg
兩 A 2兩
␻ ⫺1

共31兲

,

where

冉 冊

A 1 ⫽ 共 a⫹k ␻ i 兲 2 ⫺k 2 ␻ r2 ⫹ k⫹

冉 冊

1 2
⫹ f 共 ⫺1 兲 f 共 1 兲
2

冉 冊

⫺ k⫹

共32兲

冉 冊

A 2 ⫽ 共 a⫺k ␻ i 兲 2 ⫺k 2 ␻ r2 ⫹ k⫹

1 2
⫹ f 共 ⫺1 兲 f 共 1 兲
2

1„f 共 ⫺1 兲 ⫹ f 共 1 兲 …⫹k ␻ r „f 共 ⫺1 兲 ⫺ f 共 1 兲 …
2⫺i 共 a⫺k ␻ i 兲 „f 共 ⫺1 兲 ⫺ f 共 1 兲 ⫺2k ␻ r ….

The eigenfunctions given by Eq. 共26兲 are shown in Fig.
3. All ␻ are real in this figure. The eigenfunctions are oscillatory in (⫺1,1), because the sign of the second term of Eq.
共22兲 is positive. If ␻ is not real, ␾ 1 must be also nonreal.
Figure 4 shows a nonreal solution to Eqs. 共22兲 and 共23兲. As
we can see from this figure, the relative phase angle between
the real part and the imaginary part is about ␲ /2. This prevents both of them from connecting properly at x⫽⫾1.
IV. SUMMARY

1⫺ k⫹ „f 共 ⫺1 兲 ⫹ f 共 1 兲 …⫹k ␻ r „f 共 ⫺1 兲 ⫺ f 共 1 兲 …
2⫹i 共 a⫹k ␻ i 兲 „f 共 ⫺1 兲 ⫺ f 共 1 兲 ⫺2k ␻ r …,

共30兲

共33兲

If ␻ i ⬎0, the left-hand side of Eq. 共31兲 is greater than unity,
while the right-hand side is less than unity. Therefore, ␻ i⬎0 cannot be satisfied. If ␻ i ⬍0, the left-hand side of Eq.
共31兲 is less than unity, while the right-hand side is greaterthan unity. Therefore, ␻ i ⬍0 cannot be satisfied. Thus ␻ i⫽0, which means stability. If ␻ i ⫽0, the eigenfunctions of
the three regions ␾ I , ␾ II , and ␾ III cannot be connected properly at x⫽⫾1. We can also show that 兩 ␻ 兩 ⬎1 for the point
spectrum.

We have shown that the magnetic shear has a strong
stabilizing effect on the diocotron instability. The fluid motion parallel to the magnetic field short-circuits the charge
perturbation of the diocotron modes. The scaling parameteris s⬅ ␻ D / ␻ p . Since the time scale of the parallel motion is⬃ ␻ ⫺1
p , the condition sⰆ1 enables the parallel motion of theplasma to cancel the perturbed charge sufficiently. Typical
non-neutral plasmas in laboratories satisfy this condition.
Mathematically the last term of the eigenequation 共15兲prohibits nonreal eigenvalues, because the last term makes
the eigenfunction oscillatory. If ␻ 苸R, the relative phaseangle between the real and imaginary parts of the eigenfunction is about ␲ /2. This phase angle disables both the real andimaginary parts of the eigenfunction to satisfy the jump conditions 共24兲 and 共25兲 simultaneously.
The diocotron instability is formally equivalent to the
Kelvin–Helmholtz instability.11 The present analysis of theeffect of the parallel dynamics has, thus, a close analogy withthe study of the Kelvin–Helmholtz instability in a shearedmagnetic field14 and auroras with the longitudinal fluctuating
electric field.15,16 In Ref. 14, Idomura et al. discussed theimportance of the magnetic shear stabilization of the

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Phys. Plasmas, Vol. 8, No. 6, June 2001

Stabilization effect of magnetic shear . . .

2639

FIG. 4. A complex solution to Eqs. 共22兲 and 共23兲 for imaginary ␻. The realpart and the imaginary part of ␾ 1 have phase displacement of about ␲ /2.
This prevents both the real and imaginary parts of ␾ 1 from connecting
properly at x⫽⫾1.

Finally we note that our analysis is based on a modal
approach which, however, may not be complete for nonHermitian systems.17–19 There remains a possibility of secular algebraic behavior, although we have shown that there are
no exponentially unstable modes. This problem will be discussed elsewhere.
ACKNOWLEDGMENTS
FIG. 3. Eigenfunctions given by Eq. 共26兲. All eigenvalues are real, and nojumps at the plasma surfaces are considered in this figure. The eigenfunctions are oscillatory in the plasma region, because the sign of the second
term of Eq. 共22兲 is positive.

Kelvin–Helmholtz instability in connection with the E⫻Bzonal flow in high-temperature tokamak plasmas.14 Their
scaling parameter is ␭ De / v te 共␭ De is the Debye length and v teis the electron thermal velocity兲, and they reported that theparallel Landau resonance is effective to stabilize theKelvin–Helmholtz instability. In Refs. 15 and 16, Thompson
and Satyanarayana introduced the parallel motion of electrons into their analyses to explain the field-aligned electricfields and currents in the shear-flow region of the magnetosphere. They reported that the compressional energy of electrons that comes from the parallel dynamics is the cause ofstabilization of the Kelvin–Helmholtz instability.

The authors are obliged to Dr. F. Volponi for his useful
discussions.
This work was supported by a Grant in-Aid for Scientific
Research from the Japanese Ministry of Education, Science
and Culture No. 09308011 and No. 12780353.
1

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